Problem: A particle moves along the $x$ -axis. The function $x(t)$ gives the particle's position at any time $t\geq 0$ $x(t)=7t-3$ What is the particle's acceleration $a(t)$ at $t=6$ ? $a(6)=$
Explanation: We have a function for the particle's position, and we need to find the particle's acceleration. Acceleration is the rate of change of velocity, which is the rate of change of position. So we need to find the second derivative of $x(t)$. In other words, if $v(t)$ gives the particle's velocity and $a(t)$ gives the particle's acceleration at any time $t\geq 0$, then $a(t)=v'(t)=x''(t)$. Let's differentiate $x(t)$ to find $v(t)$ : $\begin{aligned} v(t)&=x'(t) \\\\ &=\dfrac{d}{dx}[7t-3] \\\\ &=7 \end{aligned}$ Let's differentiate $v(t)$ to find $a(t)$ : $\begin{aligned} a(t)&=v'(t) \\\\ &=\dfrac{d}{dt}[7] \\\\ &=0 \end{aligned}$ We found that the particle's acceleration is constant at $0$. In other words, the particle has a constant velocity—it neither speeds up nor slows down. In conclusion, the particle's acceleration at $t=6$ is $0$.